Brute Force Computations and Reference Solutions

In this paper, we consider the application of brute force computational techniques (BFCT) for solving computational problems in mathematical analysis and matrix algebra in floating-point computing environment. These techniques include, among others, simple matrix computations and analysis of graphs of functions. Since BFCT are based on matrix calculations the program system MATLAB® is suitable for their computer realization. The computations in this paper are done in double precision floating-point arithmetic obeying the 2019 IEEE Standard for binary floating-point calculations. One of the aims of the paper is to analyze cases when popular algorithms and software fail to produce correct answers without warning for the user. In real time control applications this may have catastrophic consequences with heavy material damage and human casualties. It is known, or suspected, that a number of man-made catastrophes such as Dharhan accident (1991), Ariane 5 launch failure (1996), Boeing 737 Max tragedies (2018, 2019) and others are due to errors in the computer software and hardware. Another application of BFCT is finding good initial guesses for known computational algorithms. Sometimes simple and fast BFCT are useful tools in solving computational problems correctly and in real time. Among particular problems considered are genuine addition of machine numbers, numerically stable computations, finding minimums of arrays, minimization of functions, solving finite equations, integration and differentiation, computing condensed and canonical forms of matrices and clarifying the concepts of least squares method in the light of the conflict Remainders vs. Errors. Usually BFCT are applied under user’s supervision which is not be possible in automatic implementation of computational methods. To implement BFCT automatically is a challenging problem in the area of artificial intelligence (AI) and of mathematical artificial intelligence (MAI) in particular. BFCT allow to reveal the underlying arithmetic (FPA, VPA, etc.) in the performance of computational algorithms. Last but not least this paper may have some tutorial value since at certain places computational algorithms and mathematical software are still taught without taking into account the properties of computational algorithms and machine arithmetic.